190 VIII DIFFERENTIAL CALCULUS

14. TAYLOR'S FORMULA

(8.14.1) Let I be an open interval in R, /, g two functions of £$\T) and
<^P)(I) respectively, (x, y) -» [x - y] a continuous bilinear mapping of E x F
into G. Then

*~ VI -
This is immediately verified by application of (8.1.4).

(8.14.2) Let I be an open interval in R, fa function of $(/\l); then, for any
pair of points a, £ in I

i

Apply (8.14.1) to the bilinear mapping (A, x) ™>Ajc and to the function
= (f — Op"V(p ~~ 0-5 an(i integrate both sides between a and £.

(8.14.3) Lef E, F be two Banach spaces, A an open subset of E, / a p-times
continuously differentiable mapping of A into F. Then, if the segment joining
x and x + t is in A, we have

f(x + 0 =/(*) + !/'(*) • r + !/'(*) • ?<2> + - - • + -_l^-/(p- D(X) - ro-"

k) stands for (t,t,..., t) (k times). In particular, for every s > 0,
& r > 0 such that for \\t\\ ^ r

+1) -/(x) —/'(*) • t - iroo • ^2> - - - - - -f

II L\ piontinuous